On the geometric side, in general, a similar fibration result as in [1]} is not satisfied in this setting. A counterexample can be found in [2]}. However we know that if \((X, \mathsf {d})\)  is compact with no singular set, and (REF ) is non-collapsed (as explained below), then \(X\)  is homeomorphic to \(M^n_i\)  for any sufficiently large \(i\) , which is proved in [3]}. This gives a geometric answer to the problem \((\star \star )\) .
